1
vote
0answers
118 views

Dalzel's integral for $\pi$ and the lemniscate constant

$\pi=\varpi_2$ can be considered as a member of the sequence of real numbers $$\varpi_m=2\int\limits_0^1\frac{dt}{\sqrt{1-t^m}},$$ and, for example, the Wallis product formula ...
76
votes
5answers
6k views

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ where the ...
1
vote
1answer
303 views

Best rational approximation in a special sense

Let $\alpha$ be an irrational number, $n\geq 1$ and $ X_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$ Now let $(x_n,y_n)$ minimize the quantity $x+y\alpha$ on $X_n$. ...
7
votes
3answers
1k views

Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer. ...
26
votes
1answer
1k views

An inequality for cosine of n

Can anyone provide a proof of the following inequality? If $n$ is a positive integer, $n\geq2$, then $$\cos(n) \leq 1 - 2^{-n}.$$ This is satisfied if $n$ is not within about $2^{-n/2}$ of a multiple ...
19
votes
3answers
2k views

When is $n/\ln(n)$ close to an integer?

As usual I expect to be critisised for "duplicating" this question. But I do not! As Gjergji immediately notified, that question was from numerology. The one I ask you here (after putting it in my ...